Intro

Coexistence is everywhere (coral reefs, tropical forests, grasslands, gut microbiota, etc.) and should be simple to obtain.

To our knowledge, there is no general model able to predict the stable coexistence of a high number of species.

One example: Hurtt & Pacala (1995):

  • Recruitment limitation
  • Recruitment limitation allows “winning-by-forfeit” which lessens the effect of competitive asymmetries
  • Random drift, unstable coexistence

Model

Environment

We consider \(n_s=w \times h\) sites on a regular grid. Each site \(i\) is characterized by a multidimensional environment defined by \(n_v\) environmental variables \(x_1, \ldots, x_{n_v}\) (\(n_v=3\) in our example). Each environmental variable is spatially auto-correlated and derived from an intrinsic conditional autoregressive model (iCAR). Environment variables have values in \([0, 1]\). Environment on a particular site can be defined with a RGB colour. Similar sites (in terms of environment) have similar RGB colours. Environment is not uniformly distributed: some environments are more frequent than others.

Species

We consider \(n_{Sp}\) species indexed on \(j\) (\(n_{Sp}=64\) in our example). Species niche is multidimensional. Species performance is optimal at one point in the multidimensional environmental space. Species performance on each site \(p_{ij}\) is computed from the Euclidean distance \(d_{ij}\) between the optimal point for species \(j\) and the environment on each site \(i\), in the multidimensional environmental space: \(p_{ij}=-d'_{ij}\), where \(d'_{ij}\) is the normalized distance (\(d'_{ij}\)=\((d_{ij}-\mu_d)/\sigma_d\)). This means that one species will outcompete all the other species in one particular environment. Similar species have similar colours.

The number of sites where each species has a higher performance is variable between species. For some species, there is no sites where they would be the most competitive.

Dynamics

Initial conditions

Each site is occupied by one individual of one species. Species are drawn at random on each site.

Mortality

The mortality rate for each individual is a function of the species performance on each site. \(\alpha\) is the mean mortality rate and \(b\) is the effect of the species performance on the mortality. In our example, we use \(\alpha=0.1\) and \(b=-0.5\). Mortality leads to vacant sites for which species will compete.

\[\theta_{ij}=\text{logit}^{-1}(\text{logit}(\alpha) + b \times p_{ij})\]

Recruitment

There is no recruitment limitation (as in Hurtt & Pacala (1995)) in our model. We assume that each species which is still present in the community has the ability to send a propagule on each vacant site. The species with the highest performance on each site outcompetes the other species and occupies the site.

Simulations

  • Number of generations: \(g=100\).
  • Number of repetitions with changing initial conditions: \(r=10\).

Result

Species coexistence

Species without suitable environment are excluded from the community. Stable coexistence of the other species.

The mean species rank (over repetitions) at the end of the simulations is correlated to the frequency of suitable habitat for the species.

Environmental filtering

Species end up occupying the sites where they have the highest performance. Community is structured in space and follows the spatial structure of the environment.

There is a decrease of the mean mortality rate with time associated to an increase of the mean species performance over all sites.

Intraspecific variability

Niche shape

Intraspecific variability emerges from the environmental variation in space: individuals perform differently depending on the environment. For example, we can plot the relationship between species performance and the first environmental variable \(x_1\), and fit a polynomial model \(p_{ij}=\beta_{0,j}+\beta_{1,j} x_{1,i}+\beta_{2,j} x_{1,i}^2 + \varepsilon_{ij}\), \(\varepsilon_{ij} \sim Normal(0, V_j)\). We obtain the classical relationships: bell shape, increasing, or decreasing relationship, depending on the position of the species on the environmental axis. Note: this relationship (showing intraspecific variability) can also be observed if the environment is unidimensional.

Individual variability

Intraspecific variability \(V_j\) emerges from environmental variables \(x_2\) and \(x_3\) (not considered in the above model). This variability can be structured at the individual scale (“individual variability”) when one individual is repeatedly observed at one site.

Discussion

Species coexistence and IV

  • Two conditions for stable coexistence of species:
    • Variable environmental conditions
    • Each species outperforms the others in a given particular environment
  • IV emerges from (i) environmental variability, (ii) unobserved environmental axis
  • IV is the result of a variable and multidimensional environment, and is not a mechanism determining species coexistence

Limitations and perspectives

  • No competition mechanism per se in the model. We cannot define an intra- or inter-specific competition. Is it a problem?
  • Resources are not explicit in the model.
  • A species abundance is limited not because of limited resources, but because it cannot dominate in all types of environment.
  • “Each species outperforms the others in a given particular environment”: How does it translate exactly in real ecosystems? In terms of growth, mortality, fecundity?
  • With such a model, it seems that “competition” between species is important to explain species coexistence (species hierarchy in a given environment), but not the “competition for resources”. For example, some species might have a high performance in low-ressource environment (eg. shade tolerant species).
  • What if resources only limited species abundance (not presence) at one site? (cf. carrying capacity).

References

Hurtt, G.C. & Pacala, S.W. (1995) The consequences of recruitment limitation: Reconciling chance, history and competitive differences between plants. Journal of Theoretical Biology, 176, 1–12.